Question

the circle shown below has a diameter of 12 centimeters. what is the approximate area of the shaded sector?

a. \\(226\text{ cm}^2\\)
b. \\(390\text{ cm}^2\\)
c. \\(97\text{ cm}^2\\)
d. \\(102\text{ cm}^2\\)

Explanation:

Find the radius of the circle

Using the Diameter of a Circle and Radius of a Circle knowledge points

$$ r = \frac{d}{2} = \frac{12}{2} = 6\text{ cm} $$

Calculate the total area of the circle

Using the Area of a Circle knowledge point

$$ A_{\text{total}} = \pi r^2 = \pi (6)^2 = 36\pi\text{ cm}^2 $$

Determine the fraction of the circle shaded

The central angle of the shaded sector is given as \(310^\circ\).

$$ \text{Fraction} = \frac{310^\circ}{360^\circ} = \frac{31}{36} $$

Calculate the area of the shaded sector

We multiply the total area by the shaded fraction.

$$ A_{\text{sector}} = \frac{31}{36} \times 36\pi = 31\pi\text{ cm}^2 $$

Approximate the final numerical value

We use the approximation \(\pi \approx 3.14159\).

$$ A_{\text{sector}} \approx 31 \times 3.14159 \approx 97.39\text{ cm}^2 $$

Answer:

• A. \(226\text{ cm}^2\)
• B. \(390\text{ cm}^2\)
• C. \(97\text{ cm}^2\) (Correct answer)
• D. \(102\text{ cm}^2\)